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Motif-based filtrations for persistent homology: A framework for graph isomorphism and property prediction

This paper introduces a computationally efficient framework using persistent homology on motif-based cycle-density filtrations to effectively distinguish non-isomorphic graphs and predict structural properties, outperforming existing topological and geometric methods in both accuracy and cost.

Original authors: Meritxell Vila-Miñana, Robert Jankowski, Aina Ferrà Marcús, Rubén Ballester, M. Ángeles Serrano, Carles Casacuberta

Published 2026-04-17

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CC BY 4.0

Original authors: Meritxell Vila-Miñana, Robert Jankowski, Aina Ferrà Marcús, Rubén Ballester, M. Ángeles Serrano, Carles Casacuberta

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery: Are these two complex networks actually the same thing, just dressed differently?

In the world of math and computer science, this is called the Graph Isomorphism Problem. A "graph" is just a bunch of dots (nodes) connected by lines (edges). Think of it like a social network, a map of subway stations, or the chemical bonds in a molecule. Two graphs are "isomorphic" if they are structurally identical—you can relabel the dots and lines of one to make it look exactly like the other, even if they look different at first glance.

The problem? For complex networks, figuring this out is incredibly hard. It's like trying to tell if two massive, tangled ball of yarn are the same by just looking at the outside. Traditional methods often get stuck, especially when the networks are highly symmetrical (like a perfect snowflake).

This paper introduces a new, clever detective tool called Persistent Homology with Motif-Based Filtrations. Here is how it works, using some everyday analogies:

1. The Old Way: Counting the "Fingerprints"

Imagine you have two identical-looking houses.

The Old Method (Degree-based): You count how many doors each house has. If both have 4 doors, you might think they are the same. But wait! Maybe one house has all 4 doors on the front, and the other has them scattered. They look the same on paper, but they aren't.
The Curvature Method: You measure the "bend" of the walls. This helps, but sometimes two very different houses can have the same wall curvature.

These methods look at the "surface" or simple statistics. They often fail when the houses are built with perfect symmetry.

2. The New Method: The "Shape-Shifting" Filter

The authors propose a new way to look at the houses. Instead of just counting doors, they look at the patterns of rooms and hallways (which they call "motifs").

The "Motif" (The Pattern): They focus on specific shapes formed by the connections:
- Triangles: Three people all knowing each other (a tight-knit group).
- Chordless Squares: Four people in a circle where no one knows the person across from them (a "hole" in the social fabric).
- Chordless Pentagons: Five people in a ring with no shortcuts.
The "Filtration" (The Filter): Imagine you have a special camera that takes a picture of the network.
1. First, you take a photo where only the Triangles are visible.
2. Then, you take a photo where Squares are visible.
3. Then Pentagons.
4. Finally, you take a photo of all of them combined.

This sequence of photos is called a filtration. It's like peeling an onion layer by layer, but instead of layers, you are revealing different types of structural patterns.

3. The "Persistent" Part: Tracking the Ghosts

Now, here is the magic trick. As you peel back these layers (from triangles to squares to pentagons), you watch how the "shape" of the network changes.

Do the holes (cycles) appear?
Do they disappear?
How long do they "survive" as you change the filter?

The authors track these "ghosts" of shapes. If two networks are truly different, their ghosts will appear and disappear at different times. If they are the same, the ghosts will dance in perfect sync.

This creates a "Persistence Diagram"—a unique fingerprint for the shape of the network.

Why is this a Big Deal?

1. It's a Master Detective for "Look-Alikes"
The paper tested this on some of the trickiest puzzles in math: Strongly Regular Graphs. These are networks so perfectly symmetrical that almost every other method fails to tell them apart.

Analogy: Imagine two identical twins wearing the same clothes. The old methods just say, "They look the same." This new method looks at the tiny scars on their knees or the way they walk, and says, "Ah! Twin A has a scar on the left knee; Twin B doesn't. They are different!"
Result: The new method distinguished these tricky graphs perfectly, while older methods failed.

2. It's a Crystal Ball for Prediction
The authors didn't just stop at telling graphs apart. They used these "ghost fingerprints" to predict properties of real-world networks (like chemical molecules).

Analogy: If you know the specific pattern of rooms in a house, you can predict if the house is likely to catch fire or if it's energy-efficient.
Result: Their method predicted things like the "diameter" (how far you have to walk to get from one end to the other) or "clustering" (how tight-knit the groups are) better than any other method, even though it was computationally cheaper.

3. It's Sensitive to Tiny Changes
If you move just one piece of furniture in a house, the flow of the house changes.

The authors showed that their method is incredibly sensitive. If you "rewire" a single connection in a network (like moving a friend from one group to another), their "ghost fingerprint" changes immediately.
This makes it great for spotting subtle changes in social networks, biological systems, or chemical structures that other methods might miss.

The Bottom Line

This paper presents a new, powerful, and efficient way to understand complex networks. Instead of just counting the dots and lines, it looks at the shapes and patterns hidden inside the connections.

Think of it as upgrading from a black-and-white sketch of a network to a 3D hologram that reveals its true structure. It solves old puzzles that were thought to be impossible and gives us a better way to predict how these networks will behave in the real world.

1. Problem Statement

The paper addresses two fundamental challenges in network science and topological data analysis (TDA):

Graph Isomorphism: Determining whether two graphs are structurally identical (isomorphic) is a computationally hard problem (GI-complete). While exact algorithms exist, they are often expensive. Furthermore, standard heuristics like the Weisfeiler-Leman (WL) test fail on highly symmetric or regular graphs (e.g., strongly regular graphs, cubic graphs).
Graph Property Prediction & Robustness: Existing methods often struggle to capture rich structural information necessary for predicting global graph properties (e.g., diameter, girth) or to remain sensitive to small structural perturbations (edge rewiring/removal) without losing discriminatory power.

The authors argue that the choice of filtration in Persistent Homology (PH) is the critical factor determining the expressivity of the method, yet this relationship is not fully understood.

2. Methodology

The authors propose a framework using Persistent Homology computed from motif-based filtrations. Instead of using standard metrics (like degree or shortest path), they assign edge weights based on the local density of specific chordless cycles (motifs).

A. The Filtration Schemes

The core innovation is the use of edge-weighting schemes derived from the densities of chordless cycles:

$e_T$ (Triangle Density): The density of triangles containing an edge.
$e_S$ (Chordless Square Density): The density of 4-cycles with no chords.
$e_P$ (Chordless Pentagon Density): The density of 5-cycles with no chords.
$e_\Sigma$ (Sum of Cycle Densities): The sum of the densities of triangles, squares, and pentagons.

These weights are used to construct a filtration where edges (and subsequently cliques/simplices) are added based on their weight values. The graph is expanded into a simplicial complex (clique complex) up to a dimension $k$ .

B. The Pipeline

Weighting: Assign weights to edges based on the chosen motif density. Nodes are weighted by the minimum incident edge weight.
Simplicial Expansion: Fill cliques (complete subgraphs) into the graph. A simplex's weight is the maximum weight of its faces.
Persistent Homology: Compute the persistence diagram (birth/death of topological features like connected components and cycles) as the filtration parameter evolves.
Comparison: Use the Bottleneck Distance ( $d_B$ ) between persistence diagrams of two graphs to determine isomorphism. If $d_B < 10^{-8}$ , the graphs are considered isomorphic.

C. Evaluation Tasks

The framework was tested on three fronts:

Isomorphism Testing: Distinguishing non-isomorphic graphs in challenging datasets (Cubic, Minimal Cayley, Strongly Regular, and the BREC benchmark).
Property Prediction: Using persistence images (vectorized persistence diagrams) as features for a Random Forest regressor to predict structural properties (diameter, girth, clustering coefficient, etc.) on the ogbg-molhiv molecular dataset.
Sensitivity Analysis: Measuring the change in bottleneck distance under edge removal and edge rewiring on random graph models (Erdős-Rényi, Barabási-Albert, Watts-Strogatz).

3. Key Contributions

Novel Filtration Design: Introduction and systematic evaluation of chordless-cycle density filtrations ( $e_T, e_S, e_P, e_\Sigma$ ) for TDA, moving beyond simple degree or curvature-based weights.
Superiority on Symmetric Graphs: Demonstrating that motif-based filtrations can perfectly distinguish non-isomorphic graphs in families where standard methods (WL test, degree-based, curvature-based) fail, particularly at low clique expansion levels ( $k=1$ ).
Computational Efficiency: Showing that these filtrations offer state-of-the-art performance with lower computational costs compared to graphlet-based methods (which require counting automorphism orbits) and curvature-based methods.
High Sensitivity: Establishing that cycle-density filtrations are highly sensitive to local structural changes, acting as early detectors of topological perturbations.

4. Key Results

A. Graph Isomorphism

Strongly Regular Graphs (SRG): These are notoriously difficult for isomorphism testing. The proposed $e_\Sigma$ and $e_P$ filtrations achieved 100% success in distinguishing non-isomorphic SRGs at clique expansion levels $k=2$ and $k=3$ , and near-perfect performance at $k=1$ . In contrast, degree-based ($nD$) and curvature-based ($eO, eF$) filtrations often failed completely at $k=1$ .
BREC Benchmark: On the BREC dataset (a standard benchmark for graph learning), the $e_\Sigma$ filtration consistently outperformed other methods across all expansion levels ( $k=1$ to $k=4$ ). It matched or exceeded the performance of the state-of-the-art Ego-distance method and the 3-WL test but with significantly lower computational complexity.
Cubic and Cayley Graphs: The motif-based filtrations successfully distinguished graphs where degree-based methods failed due to regularity.

B. Property Prediction

On the ogbg-molhiv dataset, filtrations based on cycle densities ( $e_S, e_T, e_P, e_\Sigma$ ) consistently achieved the lowest Mean Absolute Error (MAE) and highest accuracy for predicting structural properties like diameter, girth, and average clustering coefficient.
This indicates that cycle-density filtrations capture intrinsic structural information (mesoscale organization) that is invisible to degree-based or simple curvature-based descriptors.

C. Sensitivity Analysis

Edge Rewiring/Removal: The $e_\Sigma$ and clustering coefficient ($nC$) filtrations showed the highest sensitivity to structural perturbations.
Unlike degree-based filtrations ($nD$), which remained stable under degree-preserving rewiring, cycle-based filtrations detected topological changes immediately. This suggests they are excellent for detecting subtle structural shifts before macroscopic connectivity breakdown occurs.

5. Significance and Conclusion

The paper establishes cycle-density filtrations as a powerful, interpretable, and computationally tractable framework for graph analysis.

Bridging TDA and Network Science: It connects the geometric intuition of network motifs (triangles, squares, pentagons) with the topological machinery of persistent homology.
Practical Applicability: The method is scalable (linear time complexity for sparse graphs) and effective for real-world applications in chemistry (molecular structure), neuroscience, and social network analysis.
Theoretical Insight: The results suggest that the "expressivity" of persistent homology is heavily dependent on the filtration choice. By encoding mesoscale connectivity patterns (cycles), these filtrations provide a representation that bridges local descriptors (degree) and global summaries (spectra), offering a robust solution for graph comparison and learning tasks.

In summary, the authors demonstrate that motif-based filtrations are not only superior for distinguishing isomorphic graphs in difficult regimes but also serve as highly effective features for predicting graph properties and detecting structural anomalies.